Math Problem Statement

Solve each system of equations by graphing: 1. y = -2x - 2 and y = 3x - 7 2. x + y = -5 and y = (1/3)x - 2 3. y = x and y = 2x 4. 3x + 2y = -3 and 2x - 3y = -15

Solution

Let's solve these systems of equations by graphing. I will walk you through each question.

17.

y = 3x - 7$$ - Plot both lines and find the point where they intersect. - **Solution:** Set the equations equal to each other: $$-2x - 2 = 3x - 7$$ Solving: $$5x = 5 \Rightarrow x = 1$$. Substitute $$x = 1$$ into either equation: $$y = -2(1) - 2 = -4$$. **Intersection:** $$(1, -4)$$. --- **18.** $$x + y = -5 y = \frac{1}{3}x - 2$$ - Rearrange the first equation into slope-intercept form: $$y = -x - 5$$. - Now plot both lines and find the intersection point. - **Solution:** Set the equations equal: $$-x - 5 = \frac{1}{3}x - 2$$ Solving: $$-3x - 15 = x - 6 \Rightarrow -4x = 9 \Rightarrow x = -\frac{9}{4}$$. Substituting $$x = -\frac{9}{4}$$ into either equation gives $$y = -\frac{11}{4}$$. **Intersection:** $$\left(-\frac{9}{4}, -\frac{11}{4}\right)$$. --- **19.** $$y = x y = 2x$$ - Graph both lines and find where they intersect. - **Solution:** Set $$x = 2x$$. This gives $$x = 0$$. Substituting $$x = 0$$ gives $$y = 0$$. **Intersection:** $$(0, 0)$$. --- **20.** $$3x + 2y = -3 2x - 3y = -15$$ - Convert to slope-intercept form or solve algebraically. - **Solution:** Using elimination: Multiply the first equation by 3 and the second by 2: $$9x + 6y = -9 4x - 6y = -30$$ Add the equations: $$13x = -39 \Rightarrow x = -3$$. Substituting $$x = -3$$ into $$3x + 2y = -3$$: $$3(-3) + 2y = -3 \Rightarrow -9 + 2y = -3 \Rightarrow 2y = 6 \Rightarrow y = 3$$ **Intersection:** $$(-3, 3)$$. --- Do you have any questions about these solutions, or would you like further details on any step? --- Here are 5 related questions to expand your understanding: 1. How do you convert an equation from standard form to slope-intercept form? 2. What are some advantages of using elimination versus substitution? 3. How do you determine if two lines are parallel or intersect? 4. What does it mean if two equations have no solutions (graphically)? 5. How can graphing calculators help in solving systems of equations? **Tip:** When solving systems by graphing, small calculation errors can change the intersection point. Always double-check your work when substituting values.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Equations
Graphing

Formulas

y = mx + b (slope-intercept form)
Setting two equations equal to each other to find intersection points

Theorems

Intersection of lines theorem

Suitable Grade Level

Grades 8-10

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